"The insights gained and garnered by the mind in its wanderings among basic concepts are benefits that theory can provide. Theory cannot equip the mind with formulas for solving problems, nor can it mark the narrow path on which the sole solution is supposed to lie by planting a hedge of principles on either side. But it can give the mind insight into the great mass of phenomena and of their relationships, then leave it free to rise into the higher realms of action." (Carl von Clausewitz, "On War", 1832)
"No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical." (Hans Freudenthal, "The Concept and the Role of the Model in Mathematics and Natural and Social Sciences", 1961)
"The final test of a theory is its capacity to solve the problems which originated it." (George Dantzig, "Linear Programming and Extensions", 1963)
"If we deal with our problem not knowing, or pretending not to know the general theory encompassing the concrete case before us, if we tackle the problem 'with bare hands', we have a better chance to understand the scientist's attitude in general, and especially the task of the applied mathematician." (George Pólya, "Mathematical Methods in Science", 1977)
"[...] two related deficiencies have prevented real progress in understanding insight and its role in problem solving. First, we do not yet have a system of classifying problems into those in which insight occurs versus those in which it does not. However, only if we can isolate problems in which insight occurs will we be able to set on a firm base our theories of the mechanisms underlying insight. Second, formulation of such a taxonomic system requires that we agree on a definition of insight." (Robert W Weisberg, "Prolegomena to theories of insight in problem solving: a taxonomy of problems", 1995)
"Mathematics is not a matter of 'anything goes', and every mathematician is guided by explicit or unspoken assumptions as to what counts as legitimate – whether we choose to view these assumptions as the product of birth, experience, indoctrination, tradition, or philosophy. At the same time, mathematicians are primarily problem solvers and theory builders, and answer first and foremost to the internal exigencies of their subject." (Jeremy Avigad, "Methodology and Metaphysics in the Development of Dedekind’s Theory of Ideals", 2006)
"Therefore, although the notion of insight as a distinct process has a long history in the psychological study of problem solving, it might be useful at this point to refrain from using analytic and insight as theoretical terms applied a priori to problems." (Jason M Chein et al, "Working memory and insight in the nine-dot problem", Memory & Cognition 38, 2010)
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